Chapter 22: Ratio, Proportion and Proof

Ratios

• Ratios are used to compare quantities
• You can find equivalent ratios by multiplying or dividing by the same number

Simplest form

• To write a ratio in its simplest form, find an equivalent ratio with the smallest possible whole number values

Problem solved

• If you use x to represent one part of the ratio, then you can write the length of the x as x
• Then form an equation and solve it to find x

Harder questions

• If the ratio contains decimals or fractions-multiply
• If the ratio has mixed units-convert to the smaller unit

Proportion

• Two quantities are in direct proportion when both quantities increase at the same rate
• Two quantities are in inverse proportion when one quantity increases at the same rate as the other quantity decreases

Divide or multiply

• You can use common sense to work out whether to divide or multiply in proportion questions
  • 6 people build a wall in 4 days
  • 6x4=24 so 1 person could build the wall in 24 days
  • 24/8=3 so 8 people could build the wall in 3 days

Examiners report

• Show your method by writing down what you are working out at each stage of your working
• Read the question carefully-you have to write down the option which offers better value
• You won’t get the final mark if you just circle or underline your choice

Proof

• You can use algebra to prove facts about numbers
• Using algebra helps you to probe that something is true for every number
• In a proof question the working is the answer
• If the question say ‘show that’ or ‘prove that’ you need to write down every stage of your working
  • If you don’t you won’t get all the marks

Golden rule

• If you need to prove something about numbers then you always use algebra

Algebraic proof toolkit

• Even number = 2n
• Odd number = 2n+1 or 2n-1
• Multiple of 3 = 3n
• Consecutive numbers = n,n+1.n+2…
• Consecutive even numbers = 2n,2n+2,2n+4…
• Consecutive odd numbers = 2n+1,2n+3,2n+5…